We describe topological dynamics over a space by starting from a simple ODE emerging out of two coupled variables. We describe the dynamics of the evolution of points in space within the deterministic and stochastic frameworks. Historically dynamical systems were associated with celestial mechanics. The core philosophies of two kinds of dynamics emerging from Poincaré and Lyapunov are described. Smale’s contributions are highlighted. Markovian models are considered. Semi-group actions are a tool in this study. pmc J Indian Inst Sci J Indian Inst Sci 3814 phenaturepg 101716487 Journal of the Indian Institute of Science 0970-4140 0019-4964 pmc-is-collection-domain yes pmc-collection-title Springer Nature - PMC COVID-19 Collection PMC8342274 PMC8342274.1 8342274 8342274 34376929 10.1007/s41745-021-00257-x 257 1 Review Article Dynamical Systems: From Classical Mechanics and Astronomy to Modern Methods Rao Arni S. R. Srinivasa arni.rao2020@gmail.com arrao@augusta.edu 1 Arni S.R. Srinivasa Rao was born and raised in India until he obtained his PhD. He is a Professor and Director of Laboratory for Theory and Mathematical Modeling, Medical College of Georgia, Augusta, U.S.A. Keywords Topological dynamics Stochastic dynamics Evolution Mathematics Subject Classification 37-01 37D45 54H20 pmc-status-qastatus 0 pmc-status-live yes pmc-status-embargo no pmc-status-released yes pmc-prop-open-access yes pmc-prop-olf no pmc-prop-manuscript no pmc-prop-legally-suppressed no pmc-prop-has-pdf yes pmc-prop-has-supplement no pmc-prop-pdf-only no pmc-prop-suppress-copyright no pmc-prop-is-real-version no pmc-prop-is-scanned-article no pmc-prop-preprint no pmc-prop-in-epmc yes issue-copyright-statement © Indian Institute of Science 2021 Overview Dynamical systems arise in several practical real-world situations apart from classical physical systems like astronomy, mechanics, etc., The dynamics could be studied purely within a single variable, say a set X or between two interacting or coupled variables, say X and Y or between several interacting variables. Classical mechanics models do not capture much complexity and hence the assumptions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta _{X}$$\end{document} Θ X are straightforward. Suppose there is another space Y that has some influence over the values that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{t}^{x}$$\end{document} φ t x picks. Later in this section, we have mentioned a classical pendulum example where the space Y we have indicated as a magnetic field. In classical pendulum mechanics, this phenomenon of Y influencing the values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _{t}^{(x,y)}$$\end{document} φ t ( x , y ) picked from the space X can be treated as adding a magnetic field around the pendulum that influences the path of the oscillations. By changing the degree of a magnetic field the motion of a pendulum can be altered. Y acts like an external factor. \end{aligned}$$\end{document} lim n → π p x i x j ( n ) = π x j . One of the key differences between dynamics due to Lyapunov and the dynamics in the stochastic framework is that, in the former, the trajectories created by a set of initial values will be unique. In the next section, we will review classical dynamical systems explained by Poincaré and Lyapunov. We will discuss the topological dynamics due to Stephen Smale 7 as well as ergodicity results of Katok 2 . Topological Dynamics Mathematically, topological dynamics was first studied by Henri Poincaré during the early 20th century 2 . Some understanding of the dynamics of celestial objects through celestial mechanics has existed as far back as ancient Indian and ancient Greek works of literature 8 , 9 . However, Poincaré first conceptualized the idea of topological dynamics while understanding the qualitative properties of differential equations. Among the many technicalities, the ideas of homeomorphisms, topological spaces, semigroups of continuous transformations between spaces, diffeomorphisms, flows between various states of spaces, etc., played a central role in several advancements in the field. Several ideas of planetary motion, gravitational forces, solar system movements later termed celestial mechanics during the post-Copernican era were known to ancient Indian and Greek philosophers, astrologers, and mathematicians. Such formulations as in Poincaré do not change the course of the dynamics after initial values or initial points of reference that were set at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{0}$$\end{document} t 0 or at 0. Even the global stability features that were discussed earlier or the local stability features of Hartman–Grobman were purely mathematical possibilities and would perfectly fit well Because the diffeomorphism property was involved, the horseshoe can be reverted to the initial square. Two Distinct Dynamics from the Same Origin The idea of this section is to explain how the dynamics created by a system are distinct if we keep updating the system with newer information available on the trajectories. Even if these two distinct dynamical systems are generated from the same origin, we could see two or more different dynamics emerge.